Empirical Binomial Hierarchical Bayesian Modeling … Binomial Hierarchical Bayesian Modeling (EBHBM) 3/42 Overview Overview Seminar Aims R & WinBUGS Basics Bayesian Inference and

Empirical Binomial Hierarchical Bayesian Modeling (EBHBM) 1 / 42

Empirical Binomial Hierarchical BayesianModeling (EBHBM)

Using EBHBM to Determine Whether a Behavioral InterventionWorks Well for Some Participant Groups but Less so for Others

Yuelin Li, PhD.Department of Psychiatry & Behavioral Sciences

Department of Epidemiology & BiostatisticsMemorial Sloan-Kettering Cancer Center

SBM 2011, Washington DC3:15 – 6:00 PM, April 27, 2011


Example EBHBM

Expr Ctrl PosteriorSite quit N % quit N % E − C Mean Pr(E > C |Data)1 0 17 0% 2 21 10% -0.10 -0.09 0.0022 8 91 9% 6 92 7% 0.02 0.03 0.733 6 53 11% 5 54 9% 0.02 0.02 0.644 4 107 4% 5 113 4% -0.007 -0.007 0.395 1 42 2% 3 38 8% -0.06 -0.05 0.306 18 106 17% 19 120 16% 0.01 0.01 0.597 2 16 13% 0 23 0% 0.13 0.13 0.998 1 32 3% 1 27 4% -0.006 -0.006 0.469 4 29 14% 2 36 6% 0.08 0.08 0.88

44 493 9% 43 524 8%


Overview

Overview

Seminar Aims

R & WinBUGS Basics

Bayesian Inference and Conjugate Prior

Bayesian Approach in Moderation Analysis in Behavioral Research

Example 1: Smoking Cessation or Reduction in Pregnancy Trial

Gibbs Sampling

Example 2: Presurgical Smoking Cessation for Cancer Patients

EBHBM in Your Own Research

Summary


Overview

Overview

Seminar Aims

R & WinBUGS Basics




Gibbs Sampling



Summary


Overview

Overview

Seminar Aims

R & WinBUGS Basics




Gibbs Sampling



Summary


Overview

Overview

Seminar Aims

R & WinBUGS Basics




Gibbs Sampling



Summary


Overview

Overview

Seminar Aims

R & WinBUGS Basics




Gibbs Sampling



Summary


Overview

Overview

Seminar Aims

R & WinBUGS Basics




Gibbs Sampling



Summary


Overview

Overview

Seminar Aims

R & WinBUGS Basics




Gibbs Sampling



Summary


Overview

Overview

Seminar Aims

R & WinBUGS Basics




Gibbs Sampling



Summary


Overview

Overview

Seminar Aims

R & WinBUGS Basics




Gibbs Sampling



Summary


Seminar Aims

Seminar Aims

I What EBHBM can do

I What EBHBM cannot do

I How to carry out EBHBM analysis and interpret results

I Immediately apply these techniques in your own research

I Old data can be analyzed by a novel method?

I Help plan your next study?


R & WinBUGS Basics

R & WinBUGS Basics Demonstration






General Framework of Bayesian Inference (Lee, 2004)1

I Suppose we want to know the values of k unknown quantities

θ = (θ1, θ2, θ3, . . . , θk), (where k can be one or more than one)

I You have some a priori beliefs about their valuesp(θ)

I Suppose you obtain some data relevant to their valuesX = (X1,X2,X3, . . . ,Xk)

I Likelihood of datap(X|θ) = l(θ|X)

I From Bayes’ theorem we know

p(θ|X) ∝ p(θ)p(X|θ)

posterior ∝ prior× likelihood

1Chapter 2 of Lee, PM. (2004). Bayesian Statistics, Arnold, London



Simple Example: Beta-Binomial Model2

I A few days before 2004 presidential election: Kerry vs Bush

θ = θ1 = Pr(Kerry voters in OH), (k = 1)

I Beta prior: Three previous polls had been conducted. (942individuals said they would vote for Kerry and 1008 individualswould vote for Bush)

p(θ) =Γ(α + β)

Γ(α)Γ(β)θα−1(1− θ)β−1 = Be(942− 1, 1008− 1)

2Lynch, S. M. (2007). Intro to Applied Bayesian Stat & Estim Soc Sci.



Beta-Binomial Model [continued]

I Binomial likelihood data: Suppose the latest poll: 556 forKerry, 511 for Bush; Likelihood data follows a binomialdistribution

X = X1 = (556 Kerry, 511 Bush)

p(X|θ) = l(θ|X) = Binom(556,θ) =

(1067

556

)θ556(1− θ)511

I Posterior is also a Beta distribution Be(α = 1498, β = 1519)

p(θ|X) ∝ θ556(1− θ)511θ941(1− θ)1007 = θ1497(1− θ)1518

I Descriptive characteristics of a Beta distribution:mean = α

α+β , mode = α−1α+β−2 , variance= αβ

(α+β)2(α+β+1)




0.0 0.2 0.4 0.6 0.8 1.0

010

2030

40

x

dbet

a(x,

sha

pe1

= 9

41, s

hape

2 =

100

7)

Beta prior

0.40 0.45 0.50 0.55 0.60

010

2030

40

dbet

a(x,

sha

pe1

= 9

41, s

hape

2 =

100

7)

0.0 0.2 0.4 0.6 0.8 1.00

1020

3040

x

dbin

om(x

= 5

56, s

ize

= 1

067,

pro

b =

x)

* 10

67

Binomial likelihood

0.40 0.45 0.50 0.55 0.60

010

2030

40

dbin

om(x

= 5

56, s

ize

= 1

067,

pro

b =

x)

* 10

67

0.0 0.2 0.4 0.6 0.8 1.0

010

2030

40

x

dbet

a(x,

sha

pe1

= 1

497,

sha

pe2

= 1

518)

Beta posterior

0.40 0.45 0.50 0.55 0.60

010

2030

40

dbet

a(x,

sha

pe1

= 1

497,

sha

pe2

= 1

518)




95% HDR = (0.479, 0.514)

50% HDR = (0.490, 0.503)

0.46 0.48 0.50 0.52 0.54

Posterior mode: Kerry 49.7% = (1498− 1)/(1498 + 1518− 2)2004 election outcome Ohio: Kerry 48.7%, Bush 50.8%



Variety of Beta Distributions

0 < a < 1

A

0 < b < 1

B

b = 1

C

1 < b < 2

D

b = 2

E

b > 2

a = 1

F G H I J

1 < a < 2

K L M N O

a = 2

P Q R S T

a > 2



Another Beta-Binomial Example3

I Albert (2007) pp. 237-238

I Beta prior p(θ) = Be(α = 0.5, β = 0.5),

I Binomial likelihood data l(θ|X) = Binom(7 successes out of50)

I Estimate Beta posterior of p(θ|X)Beta(0.5+7, 0.5+43) with a mean of 0.5+7

0.5+7+0.5+43 = 0.14706

3Albert (2007). Bayesian Computation with R. Springer



Albert (2007) Example

0.0 0.2 0.4 0.6 0.8 1.0

02

46

8Beta prior

0.0 0.2 0.4 0.6 0.8 1.0

02

46

8

Binomial likelihood data

0.0 0.2 0.4 0.6 0.8 1.0

02

46

8

Beta posterior

I Misinformed prior beliefs can be corrected given sufficient data

I How to do this in WinBUGS?



How to Estimate Beta Posterior

Now we turn to R and WinBUGS



Theoretical vs Empirical Distributions

Histogram of Albert.bugs$sims.array[, 1, "p"]

Albert.bugs$sims.array[, 1, "p"]

Den

sity

0.1 0.2 0.3 0.4

02

46

8



Highest Density Region

0.0 0.2 0.4 0.6 0.8 1.0

02

46

8

50% HDR = (0.102, 0.167)

0.0 0.2 0.4 0.6 0.8 1.00

24

68

95% HDR = (0.058, 0.245)

I Package pscl by Simon Jackman et al.

I betaHPD(alpha = 7.5, beta = 43.5, p = 0.95)



Summary of Basic Bayesian Inference

I Bayesian ConjugacyI Important concept of “conjugacy” in Bayesian statisticsI The prior and likelihood are said to be “conjugate” when the

posterior distribution follows the same form as the priorI But in real-life problems the prior and posterior are not

necessarily conjugate

I Beta-binomial model for clear modeling of dichotomousoutcomes

I Learn new information, systematic incorporation of knowledge

I Population quantities can be changing over time

I Straightfoward assessment of model fit



We have covered enough basicsMove on to Bayesian Approach in Moderation Analysis



Example Conventional Moderation Analysis

Wickens (1989, p.78, Table 3.2)

Accept RejectMale 26 41Female 14 51

χ2df=1 = 3.88, p = 0.049

Department 1 Department 2Accept Reject Accept Reject

Male 23 16 3 25Female 7 4 7 47

Conditional independence: sex and rejection are independent givenacademic department: The Pearson X 2

(df=2) = 0.16, p = 0.92



Approaches in Moderation Analysis

I Conventional ApproachI Regression-like approach (e.g., Baron & Kenny, 1986)I First, establish an intervention effectI Next, intervention effect becomes non-significant when the

moderator is entered

I Bayesian ApproachI Directly estimate Pr(E > C |data)I Advantages (Gill, 2008)4

I Overt and clear model assumptionsI A rigorous way to make probability statementsI An ability to update these statements as evidence accrueI Missing info handled seamlesslyI Recognition that population quantities can be changing over

time rather than forever fixedI Handle hierarchical data easily

4Gill J (2008). Bayesian Methods. Chapman & Hall.



Example 1Smoking Cessation or Reduction in Pregnancy Trial

(SCRIPT)



SCRIPT Study

I NHLBI/NIH (R01HL56010) led by Richard Windsor, PhD.

I N=1,017 Medicaid-eligible pregnant women who smoke

I Intervention (E group): enhanced patient education materials

I Control (C group): standard public health care pamphlets

I Recruited from 9 Medicaid maternity care sites in Alabama

I Smoking abstinence at 60-day follow-up

I Biochemical verification by saliva cotinine < 30ng/mL



SCRIPT Data

Expr Ctrl PosteriorSite quit N % quit N % E − C Mean Pr(E > C |Data)1 0 17 0% 2 21 10% -0.10 -0.09 0.0022 8 91 9% 6 92 7% 0.02 0.03 0.733 6 53 11% 5 54 9% 0.02 0.02 0.644 4 107 4% 5 113 4% -0.007 -0.007 0.395 1 42 2% 3 38 8% -0.06 -0.05 0.306 18 106 17% 19 120 16% 0.01 0.01 0.597 2 16 13% 0 23 0% 0.13 0.13 0.998 1 32 3% 1 27 4% -0.006 -0.006 0.469 4 29 14% 2 36 6% 0.08 0.08 0.88

44 493 9% 43 524 8%



EBHBM of SCRIPT Data

I Hardin et al. (2008)5

I Model represented in a diagram

and beta distributions are conjugate distributions. By choosing a betaprior, it makes the mathematical computations of the posterior distribu-tion of interest more convenient to compute (Gelman et al. 1995). Basedon Gelman et al. (1995), this natural hierarchical setting is represented in Figure 1.

Estimates of α and β are obtained directly from the data. For each clinic,α is the number of successes (quitters) in each group and β is the numberof failures (nonquitters). Using these values for α and β, the posterior dis-tribution for the difference in the quit rates between the E and C groups isobtained.

The main interest in this study is the posterior distribution of the differ-ence between the E and C quit rates. The joint posterior distribution of theseprobabilities is given by p(δj | y) = p(y | δj)p(δj), where p(y | δj) is the like-lihood function and p(δj) is the prior distribution. To make inferences for asingle δk for clinic k, one must integrate the joint posterior distribution overall the other parameters δj(j = 1, 2, . . . , k – 1, k + 1, . . . 9) as follows:p(δk | y) = ∫ p(δ | y)dδ1dδ2 . . . dδk–1dδk+1 . . . dδ9 .

Hardin et al. / Using Bayesian Model to Analyze Data From Multisite Studies 151

Figure 1Diagram of the Hierarchical Bayesian Model

Prior distribution

Hyper parameters

βα

1θ

2θ

9θ

8θ...

...1y 2

y 8y

9y

2008 at Ebsco Electronic Journals Service (EJS) on December 29,http://erx.sagepub.comDownloaded from

5Hardin JM et al. (2008). Eval Revw; 32, 143 – 156.



Mathematical representation

I WinBUGS syntax requires clear model specification

yiE : abstainers in E per clinic, i = 1, 2, . . . , 9

yiC : abstainers in C per clinic

niE : total number of participants in E per clinic

niC : total number of participants in C per clinic

yiE ∼ dbin(θiE , niE ) yiC ∼ dbin(θiC , niC ) [likelihood]

θi ∼ Beta(αi , βi ),where αi = yi , βi = ni − yi [priors]

find δi = θiE − θiC



SCRIPT Data

Expr Ctrl PosteriorSite quit N % quit N % E − C Mean Pr(E > C |Data) Pr(E > C |Data)1 0 17 0% 2 21 10% -0.10 -0.09 0.002 0.032 8 91 9% 6 92 7% 0.02 0.03 0.73 0.803 6 53 11% 5 54 9% 0.02 0.02 0.64 0.694 4 107 4% 5 113 4% -0.007 -0.007 0.39 0.355 1 42 2% 3 38 8% -0.06 -0.05 0.30 0.046 18 106 17% 19 120 16% 0.01 0.01 0.59 0.637 2 16 13% 0 23 0% 0.13 0.13 0.99 0.998 1 32 3% 1 27 4% -0.006 -0.006 0.46 0.449 4 29 14% 2 36 6% 0.08 0.08 0.88 0.95

44 493 9% 43 524 8%

I What do you think?I Sites 7 and 9 are specialI Leadership? Other site-specific characteristics?I Resource allocation in the future?


Gibbs Sampling

How does the Gibbs sampler work?


Gibbs Sampling

WinBUGS uses the Gibbs Sampler

I Estimate a parameter vector θ = (θ1, . . . , θk)

I The joint posterior distribution of θ is [θ|data]

I Distribution typically has a complicated form

I Suppose we define a set of conditional distributions:

[θ1 | θ2, θ3, θ4, . . . , θk ,data],

[θ2 | θ1, θ3, θ4, . . . , θk ,data],

[θ3 | θ1, θ2, θ4, . . . , θk ,data],

· · ·[θk | θ1, θ2, . . . , θk−1, data].

I Gibbs sampler estimates [θ|data] by simulating from theseindividual conditional distributions


Gibbs Sampling

Now turn to R for an illustrative example of Gibbs sampler



Example 2Presurgical Smoking Cessation for Cancer Patients



Resolve Study

I Led by Jamie Ostroff, PhD., MSKCC

I NCI: R01CA90514 (PI: Ostroff)

I N=185 newly diagnosed cancer patients

I Enrolled at least 7 days before surgery

I Goal was to stop smoking before hospitalization for surgery

I Participants recruited from 7 Disease Management TeamsI Randomized individually to either

I n=89 Standard Care (NRT, Counseling)I n=96 SRS (Scheduled Reduced Smoking, NRT, Counseling)

I Primary outcome: biochemically verified 24-hour pointabstinence on the day of hospitalization for cancer surgery



Resolve Raw Data

SRS SC PosteriorSite quit N % quit N % E − C Mean Pr(E > C |Data)

Colorectal 3 8 38% 0 4 0% 0.38 . .Breast 7 14 50% 2 8 25% 0.25 . .

Urology 13 20 65% 10 18 56% 0.09 . .GYN 5 11 45% 4 11 36% 0.09 . .

Thoracic 10 25 40% 14 30 47% -0.07 . .Gastric 4 10 40% 5 9 56% -0.16 . .

Head & Neck 2 8 25% 5 9 56% -0.31 . .44 96 46% 40 89 45%



Random Effects Hierarchical Logistic Model

I Lee & Thompson (2005) Clin Trials; 2: 163 – 173

yij : smoking abstinence for person i in DMT j ,

yij ∼ Binomial(1, πij),

logit(πij) = α + βtij + uaj + ubj tij ,

where uaj , ubj ∼ N(0,Ψ),Ψ =

[σ2a σ2abσ2ab σ2b

].

I σ2a : between-cluster variance for Control, on the log odds scale

I σ2b: between-cluster variance for Treatment effect, on the logodds ratio scale



Random Effects Hierarchical Logistic Model—Results

yij ∼ Binomial(1, πij),

logit(πij) = α + βtij + uaj + ubj tij ,

where uaj , ubj ∼ N(0,Ψ),Ψ =

[σ2a σ2abσ2ab σ2b

].

α β σ2a σ2b σ2ab Odds Ratio (95% CI)

-0.299 0.072 0.5692 0.4502 -0.051 1.075 (0.425, 3.037)



Posterior Estimates

SRS SC PosteriorSite quit N % quit N % E − C Mean Pr(E > C |Data)

Colorectal 3 8 38% 0 4 0% 0.38 0.01 0.430Breast 7 14 50% 2 8 25% 0.25 0.03 0.568

Urology 13 20 65% 10 18 56% 0.09 0.10 0.795GYN 5 11 45% 4 11 36% 0.09 -0.03 0.515

Thoracic 10 25 40% 14 30 47% -0.07 -0.01 0.379Gastric 4 10 40% 5 9 56% -0.16 0.02 0.460

Head & Neck 2 8 25% 5 9 56% -0.31 0.00 0.33544 96 46% 40 89 45%



Results [continued]

●

●

●

●

●

●

●

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

Br

GMT

GYN

HNeck

Hepa_Colr

Thora

Uro

Log Odds RatioSRS > SCSRS < SC



Conclusions on the Resolve Study

I Random effects hierarchical logistic model is useful

I Posterior abstinence estimates: 43% in SC and 44% in SRSI DMT-specific intervention effects: non significant

I Seems strongly influenced by overall effectsI Small DMT groups carry little weight

I Model informs future researchI SRS intervention appears best fit Urology patientsI SRS not a good fit for lung and head & neck cancer patientsI In part because of high quit rate under Standard CareI Additional variables in a future study: doctors’ advice,

recording of doctor-patient interaction, also perhaps moregrainular stratification by doctors



Bayesian Hierarchical Models in Your Own Research



Reasons to Consider Bayesian Hierarchical Modeling

I Clustered data are common

I Treatment may work well for some participant groups/clusters

I Simultaneously fit cluster-level and individual-level variables

I Gather useful information despite relative small samples

I Help plan for the design of future studiesI Hope this seminar opens possibilities for you to explore

I New analytic approach for your old data?I New R03 applications to consider EBHBM?


Summary

Summary: Back to Seminar Aims

I What EBHBM can doI Intervention fitI Directly calculate the probability of intervention success over

control for each site/cluster/subgroupI Future resource allocation

I What EBHBM cannot doI Limited help if study lacks randomizationI Limited help if not enough sample size

I How to carry out EBHBM analysis

I How to interpret the findings

I Immediately apply these techniques in your own research


Summary

Acknowledgements

I NIDA: R21CA152074-01 (PI: Li)

I NCI: R01CA90514 (PI: Ostroff)

I NCI: T32CA009461 (PI: Ostroff)

I Clinical and Translational Science Award (NIHUL1-RR024996) to Weill Cornell Medical College

I Katherine Lee and Simon Thompson for sharing WinBUGScode

I Behavioral Research Core, MSKCCI Jamie Ostroff, PhD; Jack Burkhalter, PhD; Susan Holland, MS

Documents

Empirical Binomial Hierarchical Bayesian Modeling … Binomial Hierarchical Bayesian Modeling (EBHBM) 3/42 Overview Overview Seminar Aims R & WinBUGS Basics Bayesian Inference and